The classification schema above genus level is limited in FishBase to classes, orders, families and subfamilies ranks (the latter not yet displayed).
It has the advantage to keep it simple; it has the disadvantage to ignore some important clades in fish phylogeny like Chondrichthyes, Teleostei, Acanthomorpha, Percomorpha.

In general, it follows Nelson's Fishes of the World (2006) with the noticeable exception of Elasmobranchii and Holocephali being elevated to class rank
But at family level, FishBase may follow Eschmeyer's Catalog of Fishes that is updated more frequently to integrate new stable groupings.

Number of taxa from Orders to Species are given for what they are, a useful tool for information and database management.
The design of some analyses (so-called Biodiversity analyses) that uses these numbers must be carefully assessed before processing
(see footnote on Classification List for some critical points).

Catalog of Fishes gives also number of species,
including available names in the sense of the International Code of Zoological Nomenclature, which we don't have.
Possible discrepancies between the species and genera numbers are explained in Bailly (2010). As Catalog of Fishes is updated ahead of FishBase, the numbers given there are in general more accurate.

Wiley and Johnson published a new classification for Teleostei (2010). They propose to stabilize several hypotheses published recently or some (long) time ago by listing the known evidences (synapomorphies).
FishBase will follow this new arrangement as soon as the ichthyological community endorses it, e.g., as published in a next version of Fishes of the World.
In the meantime, to avoid confusion for non-specialists, we hold on the current one.

Note that the numbers of species given below does not take into account subspecies.